Example: A birth-death-with-immigration process

We may reinterpret this model as a representation of a process that governs the growth of a firm by interpreting a birth as the addition of a basic unit to the firm size, death as a reduction in size, again in some basic unit, and immigration as an innovation that increases the size of a firm by a unit.

for size reduction by a unit, and

for size increase by a unit, which comes either as random innovation or as random proportional growth. We assume that

The master equation is

with the initial condition that k(0) = k₀. A model with α = 0 is discussed in Cox and Miller (1965, p. 165).

7.2.1 Stationary probability distribution

Setting the left-hand side of the master equation equal to zero, and applying the detailed-balance conditions, which hold for this model because of the tree graph structure, we derive the stationary probability for k, π_k, as a negative binomial distribution

The maximum of the probability occurs at k', which satisfies

Next, we verify that the time-dependent solution of the master equation indeed approaches this steady-state solution.

7.2.2 Generatingfunction

Let the probability generating function be defined by

This equation is solved by the method of characteristics. See Hildebrand (1976, Chap. 8) for example. A brief outline of the method is in the Section A.1.

The effect of the initial condition disappears as t becomes large, because the second factor above approaches one. The mean, which is calculated as ∂G∕∂z∣_z=₁, can be shown to approach a constant α∕β at the rate e ^β'. With ^[1] In general, when a term dependent on G is absent, the equation becomes dt/1 = dz∕ h(z) = dG∕0. This equation has G = a, with some constant a, as one of the two independent solutions. The other solution is obtained from the first equality, φ(z, t) = b, say. See Cox and Miller(1965, p. 158) for the necessary relation between the two constants.

Exercise 3.

(c) Suppose that each agent is characterized by a set of K attributes, each of which takes on the value of 1 or -1. The resulting K-dimensional vector is his state vector. The distance between the states of two agents is measured by the Hamming distance, which is the number of attributes on which the two agents are different. Let P_k(t) be the probability at time t that the Hamming distance is k between two specified agents. Assume that attributes change with time in such a way that μ is the rate of change, that is, in a short time span dt, the Hamming distance changes by one with probability μ dt + o(dt). The probability is governed by the master equation

Show that the solution of this equation is

7.2.4 The cumulant-generating-function

Now, instead of solving the partial differential equation for G as we have done above, we derive ordinary differential equations to determine the means, the covariance, and the variances.

Noting that

7.3